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Title: Lectures on the Hyperreals: An Introduction to Nonstandard Analysis (Graduate Texts in Mathematics, Vol 188) by Robert Goldblatt, S. Axler, F. W. Gehring, P. R. Halmos ISBN: 0-387-98464-X Publisher: Springer Verlag Pub. Date: 01 September, 1998 Format: Hardcover Volumes: 1 List Price(USD): $64.95 |
Average Customer Rating: 5 (2 reviews)
Rating: 5
Summary: Well-written, interesting subject
Comment: This is an outstanding book. First of all, the subject matter is interesting. It is often said that the worst thing about nonstandard analysis is its name. "Nonstandard" suggests that the subject is divorced from the rest of mathematics, perhaps relying on some alternative logic. The ideas, however, are completely "standard". One way to construct the reals is via equivalence classes of Cauchy sequences. The hyperreals are also equivalence classes of sequences, modulo an ultrafilter. Hence, the sequence {1/n} is representative of a hyperreal which is positive, yet smaller than any positive real number; i.e., an infinitesimal. Inverses of infinitesimals are "unlimited". These ideas permit a very simple and intuitive development of the calculus, where the derivative becomes a linear map with infinitesimal approximation error, and the Riemann integral becomes a sum over a partition with infinitesimal mesh. All the cumbersome epsilon-delta statements are banished, making for a very clean development.
Later in the book, the author introduces the concept of a "universe". Universes are essentially structures which can encompass most of "standard" mathematics: topological spaces, measure spaces, etc. It is then shown that such universes can be embedded in larger ones: nonstandard universes. This formalizes the idea that nonstandard analysis is an extension of standard mathematics, with new and interesting objects. The notion of transfer allows one to prove sophisticated statements via simple ones; e.g., see the proof of the intermediate value theorem by partitioning the domain into subintervals of infinitesimal width.
Another good aspect of the book is the quality of the writing. Most graduate-level analysis textbooks are deliberately dense, forcing beginners to spend hours per page. In contrast, this book is very easy to read, and the pages fly. This is because the author is careful to motivate the main ideas, and to include most of the logical steps in the proofs. The exercises are also excellent, being strangely both easy and instructive, making the book valuable for self-study, which was my case.
In any first edition, there are bound to be typos. This book contains remarkably few. However, the discussion of hyperfinite summation seems flawed. The author wants to sum functions over their domains, and proposes to do this by summing over the image. The problem with this, of course, is when the function is not a bijection. For example, the sum of a function which is constant and one should count the domain. However, the sum of its image is just 1. This problem is easily fixed. Define the set of all finite sequences and the function which sums them. Transfer this. Then sum functions over finite sets by making a bijection between the domain and a finite sequence. By transfer, one obtains summation of functions over hyperfinite domains.
Another small complaint: for pedagogical reasons, the author has chosen to merely state Los's theorem (on transfer), and then illustrate its use repeatedly. Although I agree with this, after becoming familiar with transfer, I reached the point where I wanted to see the proof, which should have been included somewhere at the end of the book.
Rating: 5
Summary: more than an introduction
Comment: Most of the book on hyperreal numbers I've seen use a heavy logic formalism to treat and introduce this subject. This book introduces this concept in a very intuitive way ( which becomes more and more rigourous as the author points out different arising difficulties, and the necessity to use more sophisticated tools to avoid them) , and the first 50 pages wich give the main ideas behind the construction of these numbers can be read very easily. The historic introduction of the book by itself is a jewel. I ( a condensed matter physicist ) highly recommand this book.
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Title: Non-standard Analysis by Abraham Robinson ISBN: 0691044902 Publisher: Princeton Univ Pr Pub. Date: 08 January, 1996 List Price(USD): $31.95 |
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Title: A Primer of Infinitesimal Analysis by John L. Bell ISBN: 0521624010 Publisher: Cambridge University Press Pub. Date: 28 July, 1998 List Price(USD): $40.00 |
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Title: Abraham Robinson by Joseph Warren Dauben, Benoit B. Mandelbrot ISBN: 069105911X Publisher: Princeton Univ Pr Pub. Date: 26 May, 1998 List Price(USD): $42.50 |
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Title: The Lebesgue-Stieltjes Integral: A Practical Introduction (Undergraduate Texts in Mathematics) by Michael Carter, Bruce Van Brunt ISBN: 0387950125 Publisher: Springer Verlag Pub. Date: May, 2000 List Price(USD): $52.95 |
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Title: Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics, No 60) by V. I. Arnold, A. Weinstein, K. Vogtmann ISBN: 0387968903 Publisher: Springer Verlag Pub. Date: September, 1997 List Price(USD): $64.00 |
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